In the metric space setting,
the case \(q=p\) , \(\beta =0\)  of our main result (Theorem REF ) was
established in [1]}. The proofs in [1]} were based on potential theoretic tools
such as Harnack inequalities and the existence of capacitary potentials.
In this paper, we use a different approach which
combines and develops ideas from [3]}, [4]}, [5]}, [6]} and
is better suited to the
weighted (\(\beta \ne 0\) ) case and to exponents \(q\ge p\) .
An important feature in
the proof of Theorem REF  is the use of the so-called discrete convolution
as a capacity test function.
The definition of the discrete convolution is recalled in
Section , where we also show that
if \(u\)  belongs to the Newtonian space \(N_0^{1,p}(\Omega )\) ,
then a local maximal function of \(g_u\) 
is a \(p\) -weak upper gradient of the discrete convolution \(u_t\) .
This fact will be utilized in the proof of Theorem REF .
Theorem REF  also contains a weaker variant of
the quasiadditivity property, where instead of arbitrary sets \(E\subset \Omega \) 
the quasiadditivity is tested using only unions of Whitney balls.
